Finite Group Databasebeta

← #28560088

Subgroups — #28560088

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434133C211yesmaximal, characteristic
2#21433194C211yesmaximal, characteristic
3#21433182C211yesmaximal, characteristic
4#21423002C211yesmaximal, characteristic, complemented
5#21422965C211yesmaximal, characteristic, complemented
6#21422965C211yesmaximal, characteristic, complemented
7#21422947C211yesmaximal, characteristic, complemented
8#21422945C211yesmaximal, characteristic, complemented
9#21422912C211yesmaximal, characteristic, complemented
10C8×Dic236C211yesmaximal, characteristic, complemented
11#21422875C211yesmaximal, characteristic
12#21422567C211yesmaximal, characteristic
13#21422489C211yesmaximal, characteristic
14#21422395C211yesmaximal, characteristic
15C59×#1591C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17C4×Q8×D59C2211yescharacteristic
18#16837739C411yescharacteristic
19#16837736C411yescharacteristic
20#16837696C411yescharacteristic
21#16837695C411yescharacteristic
22#16837611C411yescharacteristic
23#16837610C411yescharacteristic
24#16837559C411yescharacteristic
25#16837556C411yescharacteristic
26#16837533C2211yescharacteristic
27#16837531—21no—
28#16837530—21no—
29#16837530—21no—
30#16837529C2211yescharacteristic
31Q8×C59⋊C8C2211yescharacteristic
32C8×Dic118C2211yescharacteristic
33C8×Dic118C2211yescharacteristic
34#16837526—21no—
35#16837526C2211yescharacteristic
36C4×Dic236C2211yescharacteristic
37C4×#11745672C2211yescharacteristic
38C4×#11745672C2211yescharacteristic
39#16837464—21no—
40#16837462—21no—
41#16837458C2211yescharacteristic
42#16837457—21no—
43#16837454—21no—
44#16837454C2211yescharacteristic
45#16837453C2211yescharacteristic
46#16837453C2211yescharacteristic
47#16837453C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837152C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54Q8⋊(C59⋊C8)C2211yescharacteristic
55#16837132C2211yescharacteristic
56#16837132C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837124C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C59×#446C2211yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×Q8⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×Q8×D59C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×#11745672C2×C411yescharacteristic
70C2×#11745672C2×C411yescharacteristic
71#14273726C2×C411yescharacteristic
72#14273725C2×C411yescharacteristic
73#14273717—21no—
74#14273707C2×C411yescharacteristic
75#14273705—21no—
76#14273703C2×C411yescharacteristic
77Q8×Dic59C2×C411yescharacteristic
78C4×Dic118—21no—
79C4×Dic118—21no—
80C4×Dic118C2311yescharacteristic
81C4×Dic118C2311yescharacteristic
82#14273692—21no—
83#14273692C2×C411yescharacteristic
84#14273684—21no—
85C4⋊C4×D59—21no—
86C4⋊C4×D59C2×C411yescharacteristic
87C42×D59—21no—
88C42×D59C2311yescharacteristic
89#14273661C2×C411yescharacteristic
90#14273660C2×C411yescharacteristic
91#14273660C2×C411yescharacteristic
92C4⋊(C59⋊C8)—21no—
93C4⋊(C59⋊C8)—21no—
94C4⋊(C59⋊C8)C2311yescharacteristic
95C4⋊(C59⋊C8)C2311yescharacteristic
96Dic59⋊C8—21no—
97Dic59⋊C8—21no—
98Dic59⋊C8—21no—
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8C2×C411yescharacteristic
102Dic59⋊C8C2×C411yescharacteristic
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273652C2×C411yescharacteristic
107#14273651C2×C411yescharacteristic
108#14273651C2×C411yescharacteristic
109#14273650C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111(C2×D59)⋊C8—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8C2×C411yescharacteristic
115(C2×D59)⋊C8C2×C411yescharacteristic
116(C2×D59)⋊C8D411yescharacteristic
117(C2×D59)⋊C8D411yescharacteristic
118#14273638—21no—
119#14273637—21no—
120#14273637—21no—
121C4×C59⋊C8—21no—
122C4×C59⋊C8C2311yescharacteristic
123C4×C59⋊C8C2311yescharacteristic
124C4×C59⋊C8C2311yescharacteristic
125C8×Dic59—21no—
126C8×Dic59—21no—
127C8×Dic59—21no—
128C8×Dic59—21no—
129C2×Q16×C59C2×C411yescharacteristic
130C4×Q8×C59C2311yescharacteristic
131C4×Q8×C59C2311yescharacteristic
132#107×C59C2×C411yescharacteristic
133C4⋊C8×C59—21no—
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#103×C59C2×C411yescharacteristic
139#97×C59—21no—
140#97×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146Q8×D59—22no—
147Q8×D59—41no—
148C2×C4×D59—21no—
149C2×C4×D59—21no—
150C2×C4×D59C22×C411yescharacteristic
151C2×C4×D59C2×D411yescharacteristic
152C2×C4×D59#4111yescharacteristic
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C411yescharacteristic
156C2×Dic118C22×C411yescharacteristic
157Q16×C59—41no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C411yescharacteristic
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C2×D411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C411yescharacteristic
168C4⋊C4×C59C22×C411yescharacteristic
169C42×C59—21no—
170C42×C59—21no—
171C42×C59C2411yescharacteristic
172#11745672—22no—
173#11745672—41no—
174(C2×D59)⋊C4—21no—
175(C2×D59)⋊C4—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59C22×C411yescharacteristic
179C4⋊Dic59C22×C411yescharacteristic
180Dic59⋊C4—21no—
181Dic59⋊C4—21no—
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59C22×C411yescharacteristic
193C4×Dic59C2×D411yescharacteristic
194C4×Dic59#4111yescharacteristic
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C411yescharacteristic
199C2×C59⋊C8C22×C411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202Dic236—22no—
203C22×D59C4×D411yescharacteristic
204Q8×C59—21no—
205Q8×C59—21no—
206Q8×C59—21no—
207Q8×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59C23×C411yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C2×C4×C59C2×#4111yescharacteristic
215C2×Dic59—21no—
216C2×Dic59—21no—
217C2×Dic59—21no—
218C2×Dic59—21no—
219C2×Dic59C4×D411yescharacteristic
220C2×Dic59C4×D411yescharacteristic
221C2×Dic59C4×D411yescharacteristic
222C4×D59—21no—
223C4×D59—21no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—41no—
227C4×D59—42no—
228C4×D59—42no—
229Dic118—21no—
230Dic118—21no—
231Dic118—21no—
232Dic118—42no—
233Dic118—42no—
234Dic118#13112yes—
235C8×C59—21no—
236C8×C59—21no—
237C8×C59—21no—
238C8×C59—21no—
239C8×C59—41no—
240C8×C59—41no—
241C8×C59—41no—
242C59⋊C8—21no—
243C59⋊C8—21no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—41no—
247C59⋊C8—41no—
248C59⋊C8—41no—
249#18032—591nomaximal, Sylow
250C22×C59C2×C4×D411yescharacteristic
251C2×D59—21no—
252C2×D59—21no—
253C2×D59—21no—
254C4×C59—21no—
255C4×C59—41no—
256C4×C59—41no—
257C4×C59—41no—
258C4×C59—41no—
259C4×C59C2×#13111yescharacteristic
260C4×C59#56811yescharacteristic
261C4×C59#57811yescharacteristic
262C4×C59#57811yescharacteristic
263Dic59—22no—
264Dic59—21no—
265Dic59—21no—
266Dic59—21no—
267Dic59—41no—
268Dic59—41no—
269Dic59—41no—
270#3104—591no—
271#2986—591no—
272#2984—591no—
273#1593—591no—
274#1591D5911yescharacteristic, complemented
275#1587—591no—
276#1587—591no—
277#1584—591no—
278#1584—591no—
279#1579—591no—
280C8×Q16—591no—
281#1572—591no—
282#1497—591no—
283#1479—591no—
284#1454—591no—
285C2×C59#293311yescharacteristic
286C2×C59#295011yescharacteristic
287C2×C59#295511yescharacteristic
288D59—41no—
289#533—591no—
290C2×C4×Q8—591no—
291#484—591no—
292#484—591no—
293#477—591no—
294#477—591no—
295#462—591no—
296#462—591no—
297#451—591no—
298#451—591no—
299#446—591no—
300#446—591no—
301#446—1181no—
302#446—1181no—
303#446—1181no—
304#446C2×D5911yescharacteristic
305#446C2×D5911yescharacteristic
306C8×Q8—591no—
307C8×Q8—591no—
308C8×Q8—591no—
309C8×Q8—591no—
310C8×Q8—1181no—
311C4×Q16—591no—
312C4×Q16—591no—
313C4×Q16—591no—
314C4×Q16C2×D5911yescharacteristic
315#433—1181no—
316#433—1181no—
317#432—591no—
318#432—1181no—
319#431—591no—
320#431—591no—
321#431—591no—
322#431—591no—
323#431—1181no—
324#335—591no—
325#335C2×D5911yescharacteristic
326#334—591no—
327#334—591no—
328Q8⋊C8—591no—
329Q8⋊C8—591no—
330Q8⋊C8—591no—
331Q8⋊C8—591no—
332Q8⋊C8—591no—
333Q8⋊C8—591no—
334Q8⋊C8C2×D5911yescharacteristic
335Q8⋊C8C2×D5911yescharacteristic
336#322—591no—
337#322—591no—
338#322—591no—
339C82C2×D5911yescharacteristic
340C59#1803211yescharacteristic, complemented, Sylow
341C22×Q8—591no—
342C2×Q16—591no—
343C2×Q16—591no—
344C2×Q16—591no—
345C2×Q16C4×D5911yescharacteristic
346C4⋊Q8—591no—
347C4⋊Q8—591no—
348#125—1181no—
349C22⋊Q8—591no—
350C22⋊Q8—591no—
351C22⋊Q8—1181no—
352C4×Q8—591no—
353C4×Q8—591no—
354C4×Q8—591no—
355C4×Q8—591no—
356C4×Q8—1181no—
357C4×Q8—1181no—
358C4×Q8—1181no—
359C4×Q8C22×D5911yescharacteristic
360C4×Q8C22×D5911yescharacteristic
361#117—1181no—
362C2×C4⋊C4—591no—
363C2×C4⋊C4—1181no—
364C2×C42—591no—
365C2×C42—1181no—
366#107—591no—
367#107—591no—
368#107—591no—
369#107C4×D5911yescharacteristic
370C4⋊C8—21no—
371C4⋊C8—21no—
372C4⋊C8—591no—
373C4⋊C8—591no—
374C4⋊C8—591no—
375C4⋊C8—591no—
376C4⋊C8—591no—
377C4⋊C8—591no—
378C4⋊C8—1181no—
379C4⋊C8—1181no—
380C4⋊C8—1181no—
381C4⋊C8—1181no—
382C4⋊C8—1181no—
383C4⋊C8—1181no—
384C4⋊C8—1181no—
385C4⋊C8C22×D5911yescharacteristic
386C4⋊C8C22×D5911yescharacteristic
387#103—591no—
388#103—591no—
389#103—591no—
390#103—591no—
391#103—591no—
392#103—591no—
393#103C4×D5911yescharacteristic
394#103C4×D5911yescharacteristic
395C22⋊C8—591no—
396C22⋊C8—591no—
397C22⋊C8—591no—
398C22⋊C8—591no—
399C22⋊C8—1181no—
400C22⋊C8—1181no—
401C22⋊C8—1181no—
402#97—21no—
403#97—21no—
404#97—1181no—
405#97—1181no—
406#97—1181no—
407C4×C8—591no—
408C4×C8—591no—
409C4×C8—591no—
410C4×C8—1181no—
411C4×C8—1181no—
412C4×C8—1181no—
413C4×C8—1181no—
414C4×C8—1181no—
415C4×C8C22×D5911yescharacteristic
416C4×C8C22×D5911yescharacteristic
417C4×C8C22×D5911yescharacteristic
418C2×Q8—591no—
419C2×Q8—591no—
420C2×Q8—1181no—
421C2×Q8—1181no—
422C2×Q8—1182no—
423C2×Q8—2361no—
424C2×Q8C2×C4×D5911yescharacteristic
425C2×Q8C2×C4×D5911yescharacteristic
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—1181no—
430C22×C4—1181no—
431Q16—41no—
432Q16—1182no—
433Q16—1182no—
434Q16—2361no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—591no—
439C2×C8—591no—
440C2×C8—591no—
441C2×C8—591no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8C2×C4×D5911yescharacteristic
446C2×C8C2×C4×D5911yescharacteristic
447C2×C8D4×D5911yescharacteristic
448C2×C8D4×D5911yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—591no—
452C4⋊C4—591no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4C2×C4×D5911yescharacteristic
464C4⋊C4C2×C4×D5911yescharacteristic
465C22⋊C4—1181no—
466C22⋊C4—1181no—
467C42—21no—
468C42—21no—
469C42—591no—
470C42—591no—
471C42—591no—
472C42—1181no—
473C42—1181no—
474C42—1181no—
475C42—1181no—
476C42C23×D5911yescharacteristic
477C23—591no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—592no—
483Q8—1181no—
484Q8—1181no—
485Q8—1181no—
486Q8—2362no—
487Q8—2362no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—591no—
493C2×C4—591no—
494C2×C4—591no—
495C2×C4—1181no—
496C2×C4—1181no—
497C2×C4—1181no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—1181no—
502C2×C4—1181no—
503C2×C4—2361no—
504C2×C4—2362no—
505C2×C4—2362no—
506C2×C4C22×C4×D5911yescharacteristic
507C2×C4C2×D4×D5911yescharacteristic
508C2×C4#41×D5911yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—21no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—1181no—
517C8—1181no—
518C8—1181no—
519C8—1181no—
520C8—2361no—
521C8—2361no—
522C8—2361no—
523C22—1181no—
524C22—1181no—
525C22—1181no—
526C22C4×D4×D5911yescharacteristic
527C4—21no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—1181no—
533C4—1181no—
534C4—1181no—
535C4—1182no—
536C4—2361no—
537C4—2361no—
538C4—2361no—
539C4#131×D5911yescharacteristic
540C4#1683816811yescharacteristic
541C4#1683821511yescharacteristic
542C4#1683821611yescharacteristic
543C2—2361no—
544C2#2143282511yescharacteristic
545C2#2143295911yescharacteristic
546C2#2143299411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1591C2
socleC22×C59C2×C4×D4
prime radicalC22×D59C4×D4
cosocleC42×C59C24
hypercenter#1591D59
hypocenterC59#18032
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21423002C2semidirect
5. #21422965C2semidirect
6. #21422965C2semidirect
7. #21422947C2semidirect
8. #21422945C2semidirect
9. #21422912C2semidirect
10. C8×Dic236C2semidirect
15. C59×#1591C2semidirect
274. #1591D59semidirect
340. C59#18032semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214341332
#214331942
#214331822
#214230022
#214229652
#214229472
#214229452
#214229122
C8×Dic2362
#214228752
#214225672
#214224892
#214223952
C59×#15912
#1803259